Let F be a vector field defined by
F: R3 → R3 / F (x, y, z) = (F1( x, y, z), F2 (x, y, z,) F3 (x, y, z,) where F1,F2,F3
have continuous partial derivatives in some region R. The divergence of F is denoted
by div F, or ∇. F, and is given by.
div (F)= ∇.F = ∂F1/∂X + ∂F2/∂Y + ∂F3/∂Z.
Divergence Properties.
1. If F(x, y, z) = (F1( x, y, z,) F2 (x, y, z,) F3 (x, y, z)) is a vector field on R3, and
F1, F2 and F3 have continuous partial derivatives of second order then
div (curl (F)) = 0.
2. If f (x, y, z) is a scalar field, the divergence of its gradient vector field
div (∇.f), is given by.
div (∇.f)= ∇.∇.f= ∂2F/∂X2 + ∂2F/∂Y2 + ∂2F/∂Z2.
An expression that is often shortened by ∇2 f, where the operator ∇2 f, is called
as the Laplace operator. This operator can also be used to a field
F(x, y, z) = (F1( x, y, z,) F2 (x, y, z,) F3 (x, y, z)) applying it to each of its
component functions, ie.
∇2 f= (∇2 f1,∇2 f2,∇2 f3)
In fluid mechanics, if the fluid velocity field is given by the
vector field F, is the div (F) = 0 is said that the fluid is incompressible.
Annex an example
EXAMPLE Let the vector field F (x, y, z) = (ex sen (y),ex cos (y), z) determines
their divergence.
Solution.
div (∇.f)= ∂/∂X(ex sen(y))+ ∂/∂y (ex cos(y)) + ∂/∂z (z)
= ex sen(y)- ex sen(y)+ 1
= 1